Boundary Value Problem for a Loaded Pseudoparabolic Equation with a Fractional Caputo Operator

Abstract

Differential equations containing fractional derivatives, for both time and spatial variables, have now begun to attract the attention of mathematicians and physicists; they are used in connection with these equations as mathematical models of various processes. The fractional derivative equation tool plays a crucial role in describing plenty of natural processes concerning physics, biology, geology, and so on. In this paper, we studied a loaded equation in relation to a spatial variable for a linear pseudoparabolic equation, with an initial and second boundary value condition (the Neumann condition), and a fractional Caputo derivative. A distinctive feature of the considered problem is that the load at the point is in the higher partial derivatives of the solution. The problem is reduced to a loaded equation with a nonlocal boundary value condition. A way to solve the considered problem is by using the method of energy inequalities, so that a priori estimates of solutions for non-local boundary value problems are obtained. To prove that this nonlocal problem is solvable, we used the method of continuation with parameters. The existence and uniqueness theorems for regular solutions are proven.

1. Introduction

In recent times, there has been significant emphasis on fractional order partial differential equations, which generalize integer order partial differential equations [1,2,3,4,5,6]. These equations have become important due to their application in various fields of science, such as physics, biology, chemistry, engineering, and control theory [7,8,9,10,11,12,13,14], as well as in the analysis and modeling of many natural science problems.

References [15,16,17,18,19,20] are devoted to the study of various local and non-local initial-boundary value problems for Sobolev type differential equations and its subclass of pseudoparabolic equations.

A significant branch of differential equation theory concerns loaded equations. The study of loaded equations traces its origins to loaded integral equations. Works by Knezer A. [21], Lichtenstein L. [22], Nazarov N.N. [23], and Gabib-Zade A.Sh. [24] are dedicated to this class of loaded equations. The importance of studying such equations was emphasized by Krylov A.N., Smirnov V.I., Tikhonov A.N., and Samarsky A.A., who provided examples of applied problems in engineering and physics that can be reduced to loaded integral equations.

A great contribution to the development of the theory of loaded differential equations has been made by Budak V.M., Iskenderov A.D., Nakhushev A.M., Kaziev V.M., and Kral A.M. [25,26,27,28]. In the review papers of Nakhushev A.M., the practical and theoretical importance of studying loaded differential equations is demonstrated through numerous examples. One of the approximate methods for solving boundary value problems for differential equations is the reduction method proposed by Nakhushev A.M., which transforms integro-differential equations into loaded differential equations. In [26], the connection between non-local problems and loaded equations is first mentioned. Bitsadze-Samarsky type nonlocal problems for Laplace and heat conduction equations are equivalently reduced to local problems for loaded differential equations.

Boundary value problems for the loaded equation with the Riemann–Liouville fractional differentiation operator were studied in the works of Gekkieva S.Kh., Kerefov M.A. [29], as well as Shevyakova O.P. [30], where the unique solvability of the problem was proven within the class of functions satisfying the Hölder condition, and the solution was constructed in an explicit form.

In addition, the controllability results of fractional dynamical systems with monotone nonlinearity and observability, with regard to linear fractional dynamical systems, were represented by the fractional differential equation used with the Caputo fractional derivative of order $\alpha \in (0,1]$; these results are given in [31,32]. In [33], the authors investigated unique solvability of generalized Caputo-type fractional boundary value problems. The existence, uniqueness, and stability analyses, concerning $\psi$-Caputo fractional derivatives, were novel findings when reported in [34].

In recent times, the considerable interest in loaded equations can be attributed to the profound and intensive research of applied problems, such as long-term forecasting, the stabilization of the surfaces of groundwater and soil moisture, problems concerning optimal ecosystem control [35,36,37,38], etc. Prominent scientists actively engaged in loaded equations include Knezer A., Lichtenstein L., Budak V.M., Iskenderov A.D., Kral A.M., Nakhushev A.M., Kozhanov A.I., Jenaliev M.T., Ramazanov M.I., Shkhanukov M.Kh. [26,36,39,40], among many other researchers.

The aim of this work is to investigate the issues of solvability for problems with non-local time initial conditions and second boundary value conditions (Neumann condition) for a linear pseudoparabolic loaded equation, with respect to spatial variables. The problem, through reduction, is transformed into a loaded equation with a non-local boundary condition (a nonlocal problem). The solvability of this non-local problem is proven using the parameter continuation method.

The solvability of a series of local and nonlocal problems for non-classical mathematical physics equations has been established using the parameter continuation method in [41,42,43,44,45,46,47,48]. Among the works dedicated to the study of issues related to the unique solvability of boundary value problems for differential equations with fractional integro-differentiation operators, we note [49,50,51,52,53].

Regarding how to solve the considered problem using the method of energy inequalities, a priori estimates of solutions to a nonlocal boundary value problem are obtained. To prove the solvability of this non-local problem, we used the method of continuation with parameters. The existence and uniqueness theorems of regular (all derivatives generalized by S.L. Sobolev are included in the equation) solutions have been proven.

2. Statement of the Problem

In a rectangle $Q_T=\{x\in (0,1);\hspace{0.17em}\hspace{0.17em}t\in [0,T]\}$, we consider the following loaded pseudoparabolic equation:

$$\begin{gathered} D_{0,t}^{\alpha }u-D_{0,t}^{\alpha }{u}_{xx}-u_{xx}+cu=f(x,t)+b_1(x,t)D_{0,t}^{\alpha }{u}_{xx}(0,t)+b_2(x,t)u_{xx}(0,t), \\ 0<x<1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0<t<T, \end{gathered}$$

(1)

with initial

$$u(x,0)=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0\le x\le 1,$$

(2)

and boundary value conditions

$$u_x(0,t)=0,\hspace{0.17em}\hspace{0.17em}{u}_x(1,t)=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0\le t\le T,$$

(3)

where $D_{0,t}^{\alpha }$ is the fractional Caputo derivative of the order $0<\alpha \le 1,$ $f(x,t),$ $b_i(x,t)\hspace{0.17em}\hspace{0.17em}(i=\hspace{0.17em}1,\hspace{0.17em}2)$ and $c$ is a constant.

3. Results

Theorem 1. 

Suppose that $f(x,t),$ $\hspace{0.17em}{f}_{xx}(x,t)\in L_2(Q_T),$ $f_x(0,t),\hspace{0.17em}$ $f_x(1,t)\in C[0,T],\hspace{0.17em}$ $\hspace{0.17em}{b}_1(x,t),\hspace{0.17em}$ $b_2(x,t)\in C^2(Q_T)$. Moreover, suppose that the following conditions are true:

$$1-\frac{3}{{\epsilon }_1}\left(b_{10}^2(t)+b_{11}^2(t)\right)-\frac{{\overline{b}}_{11}^2+{\overline{b}}_{22}^2}{2c{\epsilon }_1}-4c{\epsilon }_1-3{\overline{b}}_{11}^2\ge K_0>0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}t\in [0,T],$$

(4)

where ${\overline{b}}_{11}=\underset{(x,t)\in Q_T}{\max }{b}_{1xx}(x,t),$ ${\overline{b}}_{22}=\underset{(x,t)\in Q_T}{\max }{b}_{2xx}(x,t),$ $b_{10}(t)=b_{1x}(0,t),$ $b_{11}(t)=b_{1x}(1,t),$ $0<{\epsilon }_1<\frac{1-3{\overline{b}}_{11}^2}{4c}$.

Then, there exists a solution $u(x,t)$ of the nonlocal problem (1)–(3) such that $u,\hspace{0.17em}{D}_{0,t}^{\alpha }u,\hspace{0.17em}{D}_{0,t}^{\alpha }{u}_{xx},\hspace{0.17em}{u}_{xx}\in L_{\infty }(0,T;\hspace{0.17em}{L}_2(\Omega ))$.

We will refer to the obtained solution as the regular solution, since it satisfies Equation (1) almost everywhere in $Q_{\mathrm{T}}$, and the initial and boundary values are taken in the usual sense.

Note, that for the validity of the statement of Theorem 1, condition (4) must be fulfilled. Below is an example for the coefficients $b_1(x,t)$, $b_2(x,t)$ of Equation (1), which satisfy condition (4) of Theorem 1.

For example, let $b_1(x,t)=B_1(t)$, $b_2(x,t)=B_2(t)$, where $B_1(t)$, $B_2(t)$ are twice differentiable functions in $0\le t\le T$, and $c$ is an arbitrary positive number.

Then, after conducting the necessary calculations, we have the following:

$$b_{10}(t)=b_{1x}(0,t)=0,$$

$$b_{11}(t)=b_{1x}(1,t)=0,$$

$${\overline{b}}_{11}={\overline{b}}_{22}=0.$$

In this case, condition (4) can be represented in the following form:

$$1-4c{\epsilon }_1\ge K_0\mathrm{or}0<{\epsilon }_1<\frac{1}{4c}$$

This example illustrates the non-emptiness of the class in Equation (1), for which Theorem 1 is true.

Proof. 

Differentiate Equation (1) twice by $x$, as follows:

$$D_{0,t}^{\alpha }{u}_{xx}-D_{0,t}^{\alpha }{u}_{xxxx}-u_{xxxx}+c{u}_{xx}=f_{xx}(x,t)+b_{1xx}(x,t)D_{0,t}^{\alpha }{u}_{xx}(0,t)+b_{2xx}(x,t)u_{xx}(0,t).$$

Introducing the notation $v(x,t)=u_{xx}(x,t)$, we may rewrite the last relationship in the following form:

$$\begin{array}{l}{D}_{0,t}^{\alpha }v(x,t)-D_{0,t}^{\alpha }{v}_{xx}(x,t)-v_{xx}(x,t)+cv(x,t)\\ =f_{xx}(x,t)+b_{1xx}(x,t)D_{0,t}^{\alpha }v(0,t)+b_{2xx}(x,t)v(0,t).\end{array}$$

(5)

Now, we differentiate Equation (1) by $x$ and use condition (3), as follows:

$$\begin{array}{l}{D}_{0,t}^{\alpha }{u}_x(0,t)-D_{0,t}^{\alpha }{u}_{xxx}(0,t)-u_{xxx}(0,t)+c{u}_x(0,t)\\ =f_x(0,t)+b_{1x}(0,t)D_{0,t}^{\alpha }{u}_{xx}(0,t)+b_{2x}(0,t)u_{xx}(0,t),\end{array}$$

$$\begin{array}{l}{D}_{0,t}^{\alpha }{u}_x(1,t)-D_{0,t}^{\alpha }{u}_{xxx}(1,t)-u_{xxx}(1,t)+c{u}_x(1,t)\\ =f_x(1,t)+b_{1x}(1,t)D_{0,t}^{\alpha }{u}_{xx}(1,t)+b_{2x}(1,t)u_{xx}(0,t).\end{array}$$

We introduce the following notations, as follows:

$$f_0(t)=f_x(0,t),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{f}_1(t)=f_x(1,t),$$

$$b_{10}(t)=b_{1x}(0,t),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{b}_{20}(t)=b_{2x}(0,t),$$

$$b_{11}(t)=b_{1x}(1,t),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{b}_{21}(t)=b_{2x}(1,t).$$

Taking the entered notation into account, we may rewrite these identities as follows:

$$\begin{array}{l}{D}_{0,t}^{\alpha }{v}_x(0,t)+v_x(0,t)=-f_0(t)-b_{10}(t)D_{0,t}^{\alpha }v(0,t)-b_{20}(t)v(0,t).\\ \hspace{0.17em}{D}_{0,t}^{\alpha }{v}_x(1,t)+v_x(1,t)=-f_1(t)-b_{11}(t)D_{0,t}^{\alpha }v(0,t)-b_{21}(t)v(0,t).\end{array}$$

(6)

Define the initial condition from condition (2), as follows:

$$v(x,0)=0.$$

(7)

Next, we use the following known inequalities [47]:

$$\upsilon (y,t)\le {\delta }_0{\displaystyle \underset{0}{\overset{1}{\int }}{\upsilon }_x^2(x,t)dx+\frac{4}{{\delta }_0}{\displaystyle \underset{0}{\overset{1}{\int }}{\upsilon }^2(x,t)dx}},$$

$${\left|D_{0,t}^{\alpha }\upsilon (y,t)\right|}^2\le {\delta }_0{\displaystyle \underset{0}{\overset{1}{\int }}{\left|D_{0,t}^{\alpha }{\upsilon }_x(x,t)\right|}^{2}dx+\frac{4}{{\delta }_0}}{\displaystyle \underset{0}{\overset{1}{\int }}{\left|D_{0,t}^{\alpha }\upsilon (x,t)\right|}^{2}dx},$$

(8)

$${\left|D_{0,t}^{\alpha }{\upsilon }_x(y,t)\right|}^2\le {\delta }_0{\displaystyle \underset{0}{\overset{1}{\int }}{\left|D_{0,t}^{\alpha }{\upsilon }_{xx}(x,t)\right|}^{2}dx+\frac{4}{{\delta }_0}}{\displaystyle \underset{0}{\overset{1}{\int }}{\left|D_{0,t}^{\alpha }{\upsilon }_x(x,t)\right|}^{2}dx},$$

Multiply both parts of (5) by $D_{0,t}^{\alpha }v-D_{0,t}^{\alpha }{v}_{xx}-v_{xx}$ and integrate over $x$ from $0$ to $1$, as follows:

$$\begin{array}{l}{\displaystyle \underset{0}{\overset{1}{\int }}\left[D_{0,t}^{\alpha }v-D_{0,t}^{\alpha }{v}_{xx}-v_{xx}+cv\right]}(D_{0,t}^{\alpha }v-D_{0,t}^{\alpha }{v}_{xx}-v_{xx})dx\\ ={\displaystyle \underset{0}{\overset{1}{\int }}{f}_{xx}(D_{0,t}^{\alpha }v-D_{0,t}^{\alpha }{v}_{xx}-v_{xx})dx+{\displaystyle \underset{0}{\overset{1}{\int }}\left[b_{1xx}{D}_{0,t}^{\alpha }v(0,t)+b_{2xx}v(0,t)\right](D_{0,t}^{\alpha }v-D_{0,t}^{\alpha }{v}_{xx}-v_{xx})dx}}.\end{array}$$

(9)

By using the method of integration in parts, and by substituting the transformed integrals into identity (9), we obtain the following inequality:

$$\begin{array}{l}{\Vert D_{0,t}^{\alpha }v\Vert }_{2,\Omega }^2+2{\Vert D_{0,t}^{\alpha }{v}_x\Vert }_{2,\Omega }^2+{\Vert D_{0,t}^{\alpha }{v}_{xx}\Vert }^2+{\Vert v_{xx}\Vert }^2+D_{0,t}^{\alpha }\left({\Vert v_{xx}\Vert }^2+(1+c){\Vert v_x\Vert }^2\right)+c{\Vert v_x\Vert }^2\\ \le {\displaystyle \underset{0}{\overset{1}{\int }}{f}_{xx}\cdot D_{0,t}^{\alpha }vdx}+{\displaystyle \underset{0}{\overset{1}{\int }}{b}_{1xx}(x,t)D_{0,t}^{\alpha }v(0,t)\cdot D_{0,t}^{\alpha }vdx}+{\displaystyle \underset{0}{\overset{1}{\int }}{b}_{2xx}(x,t)v(0,t)\cdot D_{0,t}^{\alpha }vdx}\\ -{\displaystyle \underset{0}{\overset{1}{\int }}{f}_{xx}{D}_{0,t}^{\alpha }{v}_{xx}dx-{\displaystyle \underset{0}{\overset{1}{\int }}\left[b_{1xx}{D}_{0,t}^{\alpha }v(0,t)+b_{2xx}v(0,t)\right]D_{0,t}^{\alpha }{v}_{xx}dx}}\\ -{\displaystyle \underset{0}{\overset{1}{\int }}{f}_{xx}{v}_{xx}dx-{\displaystyle \underset{0}{\overset{1}{\int }}\left[b_{1xx}{D}_{0,t}^{\alpha }v(0,t)+b_{2xx}v(0,t)\right]v_{xx}dx}}\\ -cv(1,t)\left[f_1(t)+b_{11}(t)D_{0,t}^{\alpha }v(0,t)+b_{21}(t)v(0,t)\right]\\ +cv(0,t)\left[f_0(t)+b_{10}(t)D_{0,t}^{\alpha }v(0,t)+b_{20}(t)v(0,t)\right].\end{array}$$

(10)

Let us estimate the right side of inequality (10). By using inequality (8) and Young’s inequality [41], we obtain the following:

$$\begin{array}{l}\left|-cv(1,t)\left(f_1(t)+b_{11}(t)D_{0,t}^{\alpha }v(0,t)+b_{21}(t)v(0,t)\right)\right|\\ \frac{c{\epsilon }_2}{2}{\left|D_{0,t}^{\alpha }v(1,t)\right|}^2+\frac{c}{2{\epsilon }_2}{\left(f_1(t)+b_{11}(t)D_{0,t}^{\alpha }v(0,t)+b_{21}(t)v(0,t)\right)}^2\\ \le \frac{c{\epsilon }_2}{2}{\left|D_{0,t}^{\alpha }v(1,t)\right|}^2+\frac{3c}{2{\epsilon }_2}\left(f_1^2(t)+b_{11}^2(t)|D_{0,t}^{\alpha }v(0,t){|}^2+b_{21}^2(t)|v(0,t){|}^2\right)\\ \le \left(\frac{c{\epsilon }_2{\delta }_0}{2}+\frac{3c{\delta }_0}{2{\epsilon }_2}{b}_{11}^2(t)\right){\displaystyle \underset{0}{\overset{1}{\int }}{\left|D_{0,t}^{\alpha }{v}_x(x,t)\right|}^{2}dx}+\left(\frac{2c{\epsilon }_2}{{\delta }_0}+\frac{6c{b}_{11}^2(t)}{{\epsilon }_2{\delta }_0}\right){\displaystyle \underset{0}{\overset{1}{\int }}{\left|D_{0,t}^{\alpha }v(x,t)\right|}^{2}dx}\\ +\frac{3c{\delta }_0}{2{\epsilon }_2}{b}_{21}^2(t){\displaystyle \underset{0}{\overset{1}{\int }}{\left|v_x(x,t)\right|}^{2}dx}+\frac{6c{b}_{21}^2(t)}{{\epsilon }_2{\delta }_0}{\displaystyle \underset{0}{\overset{1}{\int }}{\left|v(x,t)\right|}^{2}dx}+\frac{3c}{2{\epsilon }_2}{f}_1^2(t),\end{array}$$

$$\begin{array}{l}\left|-cv(0,t)\left(f_0(t)+b_{10}(t)D_{0,t}^{\alpha }v(0,t)+b_{20}(t)v(0,t)\right)\right|\\ \le \left(\frac{{\epsilon }_{2}c{\delta }_0}{2}+\frac{3c{\delta }_0}{2{\epsilon }_2}{b}_{10}^2(t)\right){\displaystyle \underset{0}{\overset{1}{\int }}{\left|D_{0,t}^{\alpha }{v}_x(x,t)\right|}^{2}dx}\hspace{0.17em}+\left(\frac{2c{\epsilon }_2}{{\delta }_0}+\frac{6c{b}_{10}^2(t)}{{\epsilon }_2{\delta }_0}\right){\displaystyle \underset{0}{\overset{1}{\int }}{\left|D_{0,t}^{\alpha }v(x,t)\right|}^{2}dx}\\ +\frac{3c{\delta }_0}{2{\epsilon }_2}{b}_{20}^2(t){\displaystyle \underset{0}{\overset{1}{\int }}{\left|v_x(x,t)\right|}^{2}dx}+\frac{6c{b}_{20}^2(t)}{{\epsilon }_2{\delta }_0}{\displaystyle \underset{0}{\overset{1}{\int }}{\left|v(x,t)\right|}^{2}dx}+\frac{3c}{2{\epsilon }_2}{f}_0^2(t),\end{array}$$

$$\left|{\displaystyle \underset{0}{\overset{1}{\int }}{f}_{xx}{D}_{0,t}^{\alpha }vdx}\right|\le \frac{1}{4{\epsilon }_3}{\Vert f_{xx}\Vert }^2+{\epsilon }_3{\Vert D_{0,t}^{\alpha }v\Vert }^2,$$

$$\hspace{0.17em}\left|b_{1xx}(x,t)D_{0,t}^{\alpha }v(0,t)\cdot D_{0,t}^{\alpha }vdx\right|\le \frac{{\overline{b}}_{11}^2}{4{\epsilon }_4}{\displaystyle \underset{0}{\overset{1}{\int }}{\left|D_{0,t}^{\alpha }v\right|}^{2}dx+{\epsilon }_4{\delta }_0{\displaystyle \underset{0}{\overset{1}{\int }}{\left|D_{0,t}^{\alpha }{v}_x\right|}^{2}dx+\frac{4{\epsilon }_4}{{\delta }_0}{\displaystyle \underset{0}{\overset{1}{\int }}|D_{0,t}^{\alpha }v{|}^{2}dx}}},$$

$$\left|b_{2xx}(x,t)v(0,t)\cdot D_{0,t}^{\alpha }vdx\right|\le \frac{{\overline{b}}_{22}^2}{4{\epsilon }_4}{\int }_0^1{\left|D_{0,t}^{\alpha }v\right|}^{2}dx+{\epsilon }_4{\delta }_0{\int }_0^1{\left|v_x\right|}^{2}dx+\frac{4{\epsilon }_4}{{\delta }_0}{\int }_0^1|v{|}^{2}dx,$$

$$\left|{\int }_0^1{f}_{xx}{D}_{0,t}^{\alpha }{v}_{xx}dx\right|\le \frac{1}{4{\epsilon }_5}{∥f_{xx}∥}^2+{\epsilon }_5{∥D_{0,t}^{\alpha }{v}_{xx}∥}^2,$$

$$\left|b_{1xx}(x,t)D_{0,t}^{\alpha }v(0,t)\cdot D_{0,t}^{\alpha }{v}_{xx}dx\right|\le {\epsilon }_6{\displaystyle \underset{0}{\overset{1}{\int }}{\left|D_{0,t}^{\alpha }{v}_{xx}\right|}^{2}dx+\frac{{\overline{b}}_{11}^2{\delta }_0}{4{\epsilon }_6}{\displaystyle \underset{0}{\overset{1}{\int }}{\left|D_{0,t}^{\alpha }{v}_x\right|}^{2}dx+\frac{{\overline{b}}_{11}^2}{{\epsilon }_6{\delta }_0}{\displaystyle \underset{0}{\overset{1}{\int }}|D_{0,t}^{\alpha }v{|}^{2}dx}}},$$

$$\left|b_{2xx}(x,t)v(0,t)\cdot D_{0,t}^{\alpha }{v}_{xx}dx\right|\le {\epsilon }_7{\displaystyle \underset{0}{\overset{1}{\int }}{\left|D_{0,t}^{\alpha }{v}_{xx}\right|}^{2}dx+\frac{{\overline{b}}_{22}^2{\delta }_0}{4{\epsilon }_7}{\displaystyle \underset{0}{\overset{1}{\int }}{\left|v_x\right|}^{2}dx+\frac{{\overline{b}}_{22}^2}{{\epsilon }_7{\delta }_0}{\displaystyle \underset{0}{\overset{1}{\int }}|v{|}^{2}dx}}},$$

$$\left|{\displaystyle \underset{0}{\overset{1}{\int }}{f}_{xx}{v}_{xx}dx}\right|\le \frac{1}{2}{\Vert f_{xx}\Vert }^2+\frac{1}{2}{\Vert v_{xx}\Vert }^2,$$

$$\hspace{0.17em}\left|b_{1xx}(x,t)D_{0,t}^{\alpha }v(0,t)\cdot v_{xx}dx\right|\le \frac{{\overline{b}}_{11}^2{\delta }_0}{4{\epsilon }_8}{\displaystyle \underset{0}{\overset{1}{\int }}{\left|v_{xx}\right|}^{2}dx+{\epsilon }_8{\delta }_0{\displaystyle \underset{0}{\overset{1}{\int }}{\left|D_{0,t}^{\alpha }{v}_x\right|}^{2}dx+\frac{4{\epsilon }_8}{{\delta }_0}{\displaystyle \underset{0}{\overset{1}{\int }}|D_{0,t}^{\alpha }v{|}^{2}dx}}},$$

$$\left|b_{2xx}(x,t)v(0,t)\cdot v_{xx}dx\right|\le \frac{{\overline{b}}_{22}^2}{4{\epsilon }_9}{\displaystyle \underset{0}{\overset{1}{\int }}{\left|v_{xx}\right|}^{2}dx+{\epsilon }_9{\delta }_0{\displaystyle \underset{0}{\overset{1}{\int }}{\left|v_x\right|}^{2}dx+\frac{4{\epsilon }_9}{{\epsilon }_9}{\displaystyle \underset{0}{\overset{1}{\int }}|v{|}^{2}dx}}}.$$

Hence, inequality (10) can be written as follows:

$$\begin{array}{l}\left(1+\frac{6c}{{\epsilon }_2{\delta }_0}\left(b_{10}^2(t)+b_{11}^2(t)\right)-{\epsilon }_3-\frac{{\overline{b}}_{11}^2+{\overline{b}}_{22}^2}{4{\epsilon }_4}-\frac{4{\epsilon }_4}{{\delta }_0}-\frac{{\overline{b}}_{11}^2}{4{\epsilon }_6{\delta }_0}-\frac{4{\epsilon }_8}{{\delta }_0}\right){∥D_{0,t}^{\alpha }v∥}_{2,\mathrm{\Omega }}^2\\ +\left(2-\frac{3{\delta }_{0}c}{2{\epsilon }_2}\left(b_{10}^2(t)+b_{11}^2(t)\right)-{\epsilon }_4{\delta }_0-\frac{{\overline{b}}_{11}^2{\delta }_0}{4{\epsilon }_6}-{\epsilon }_8{\delta }_0\right){\Vert D_{0,t}^{\alpha }{v}_x\Vert }_{2,\Omega }^2+\frac{1}{2}{\Vert v_{xx}\Vert }^2\\ +(1-{\epsilon }_5-{\epsilon }_6-{\epsilon }_7){\Vert D_{0,t}^{\alpha }{v}_{xx}\Vert }^2+\frac{1}{2}{D}_{0,t}^{\alpha }\left({\Vert v_{xx}\Vert }^2+(1+c){\Vert v_x\Vert }^2+{\Vert v\Vert }^2\right)+c{\Vert v_x\Vert }^2\\ \le \left(\frac{3{\delta }_{0}c}{2{\epsilon }_2}\left(b_{20}^2(t)+b_{21}^2(t)\right)+(c{\epsilon }_2+{\epsilon }_4+{\epsilon }_9){\delta }_0+\frac{{\overline{b}}_{22}^2{\delta }_0}{4{\epsilon }_7}\right){\Vert v_x\Vert }^2\\ +\left(\frac{6c}{{\epsilon }_2{\delta }_0}\left(b_{20}^2(t)+b_{21}^2(t)\right)+\frac{4(c{\epsilon }_2+{\epsilon }_4+{\epsilon }_9)}{{\delta }_0}+\frac{{\overline{b}}_{22}^2}{{\epsilon }_7{\delta }_0}\right){\Vert v\Vert }^2\\ +\frac{1}{4}\left(\frac{{\overline{b}}_{11}^2}{{\epsilon }_8}+\frac{{\overline{b}}_{22}^2}{{\epsilon }_9}\right){\Vert v_{xx}\Vert }^2+\frac{3c}{2{\epsilon }_2}\left(f_0^2(t)+f_1^2(t)\right)+\frac{1}{4}\left(2+\frac{1}{{\epsilon }_3}+\frac{1}{{\epsilon }_5}\right){\Vert f_{xx}\Vert }^2.\end{array}$$

Supposing that ${\epsilon }_5={\epsilon }_6={\epsilon }_7=\frac{1}{6},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\epsilon }_3=\frac{1}{2},\hspace{0.17em}\hspace{0.17em}{\epsilon }_4={\epsilon }_6={\epsilon }_8={\epsilon }_2=c{\epsilon }_1,\hspace{0.17em}\hspace{0.17em}{\delta }_0=4$, we obtain the following:

$$\begin{array}{l}\frac{1}{2}\left(1-\frac{3}{{\epsilon }_1}\left(b_{10}^2(t)+b_{11}^2(t)\right)-\frac{{\overline{b}}_{11}^2+{\overline{b}}_{22}^2}{2c{\epsilon }_1}-4c{\epsilon }_1-3{\overline{b_{11}}}^2\right){\Vert D_{0,t}^{\alpha }v\Vert }_{2,\Omega }^2\\ +2\left(1-\frac{3}{{\epsilon }_1}\left(b_{10}^2(t)+b_{11}^2(t)\right)-\frac{{\overline{b}}_{11}^2+{\overline{b}}_{22}^2}{2c{\epsilon }_1}-4c{\epsilon }_1-3{\overline{b}}_{11}^2\right){\Vert D_{0,t}^{\alpha }{v}_x\Vert }_{2,\Omega }^2+\frac{1}{2}{\Vert v_{xx}\Vert }^2\\ +\frac{1}{2}{\Vert D_{0,t}^{\alpha }{v}_{xx}\Vert }^2+\frac{1}{2}{D}_{0,t}^{\alpha }\left({\Vert v_{xx}\Vert }^2+(1+c){\Vert v_x\Vert }^2+{\Vert v\Vert }^2\right)+c{\Vert v_x\Vert }^2\le \\ \le \left(\frac{3{\delta }_{0}c}{2{\epsilon }_2}\left(b_{20}^2(t)+b_{21}^2(t)\right)+(c{\epsilon }_2+{\epsilon }_4+{\epsilon }_9){\delta }_0+\frac{{\overline{b}}_{22}^2{\delta }_0}{4{\epsilon }_7}\right){\Vert v_x\Vert }^2+\\ +\left(\frac{6c}{{\epsilon }_2{\delta }_0}\left(b_{20}^2(t)+b_{21}^2(t)\right)+\frac{4(c{\epsilon }_2+{\epsilon }_4+{\epsilon }_9)}{{\delta }_0}+\frac{{\overline{b}}_{22}^2}{{\epsilon }_7{\delta }_0}\right){\Vert v\Vert }^2+\\ +\frac{1}{4}\left(\frac{{\overline{b}}_{11}^2}{{\epsilon }_8}+\frac{{\overline{b}}_{22}^2}{{\epsilon }_9}\right){\Vert v_{xx}\Vert }^2+\frac{3c}{2{\epsilon }_2}\left(f_0^2(t)+f_1^2(t)\right)+\frac{1}{4}\left(2+\frac{1}{{\epsilon }_3}+\frac{1}{{\epsilon }_5}\right){\Vert f_{xx}\Vert }^2.\end{array}$$

When the conditions of the theorem are true, we obtain the following inequality:

$$\begin{array}{l}{D}_{0,t}^{\alpha }\left({\Vert v_{xx}\Vert }^2+(1+c){\Vert v_x\Vert }^2+{\Vert v\Vert }^2\right)+C_0\left({\Vert v_{xx}\Vert }^2+(1+c){\Vert v_x\Vert }^2+{\Vert v\Vert }^2\right)\\ \le C_1+C_2\left({\Vert v_{xx}\Vert }^2+(1+c){\Vert v_x\Vert }^2+{\Vert v\Vert }^2\right).\end{array}$$

Lemma 1. 

(Gronwall–Bellman) [54]. Let $u(t)\ge 0$ and $f(t)\ge 0$ for $t\ge t_0$ and $u(t)$, and for $t\ge t_0$ so that the following inequality is fulfilled:

$$u(t)\le c+{\displaystyle \underset{t_0}{\overset{t}{\int }}f(t_1)u(t_1)d{t}_1},$$

where $c$ is a positive constant. In this case for $t\ge t_0$ we have

$$u(t)\le c\exp {\displaystyle \underset{t_0}{\overset{t}{\int }}f(t_1)d{t}_1}.$$

Using Lemma 1, we obtain the following estimate:

$$\underset{t\in (0,T)}{vrai\max }\left({\Vert v_{xx}\Vert }^2+{\Vert v_x\Vert }^2+{\Vert v\Vert }^2\right)\le C_3\mathrm{for}\mathrm{all}0\le t\le T<\infty .$$

(11)

From relation (9), taking the condition of the theorem and estimate (11) into account, we receive another estimate, as follows:

$$\underset{t\in (0,T)}{vrai\max }\left({\Vert D_{0,t}^{\alpha }v\Vert }_{2,\Omega }^2+{\Vert D_{0,t}^{\alpha }{v}_x\Vert }_{2,\Omega }^2+{\Vert D_{0,t}^{\alpha }{v}_{xx}\Vert }^2\right)\le C_4\mathrm{for}\mathrm{all}0\le t\le T<\infty .$$

(12)

Denote

$$V_2^{\alpha }\left(Q_T\right)=\left\{u:u,\hspace{0.17em}{D}_{0,t}^{\alpha }u,\hspace{0.17em}{u}_x,\hspace{0.17em}{D}_{0,t}^{\alpha }{u}_x,\hspace{0.17em}{u}_{xx},\hspace{0.17em}{D}_{0,t}^{\alpha }{u}_{xx}\in L_{\infty }\left(0,T;\hspace{0.17em}{L}_2(\Omega )\right)\right\}.$$

The solvability of problems (5)–(7) is proven using the continuation method with parameters. □

Consider a one-parameter family of boundary value problems. Find a function $v(x,t)$, which is the solution of the following equation in rectangle $Q_T$:

$$\begin{array}{l}{L}_{\lambda }v\equiv D_{0,t}^{\alpha }v-D_{0,t}^{\alpha }{v}_{xx}-v_{xx}+cv(x,t)\\ =f_{xx}(x,t)+\lambda \left[b_{1xx}(x,t)D_{0,t}^{\alpha }v(0,t)+b_{2xx}(x,t)v(0,t)\right],\end{array}$$

(13)

with the boundary value condition

$$\begin{array}{l}{D}_{0,t}^{\alpha }{v}_x(0,t)+v_x(0,t)=-\lambda \left[f_0(t)+b_{10}(t)D_{0,t}^{\alpha }v(0,t)+b_{20}(t)v(0,t)\right],\\ D_{0,t}^{\alpha }{v}_x(1,t)+v_x(1,t)=-\lambda \left[f_1(t)+b_{11}(t)D_{0,t}^{\alpha }v(1,t)+b_{21}(t)v(1,t)\right],\end{array}$$

(14)

and initial condition

$$v(x,0)=0.$$

(15)

Using $A_{\lambda }=\left\{\lambda \in [0,1]\right\}$, denote the set of all numbers of segment $[0,1]$, for which the problems (13)–(15) are unambiguously solvable in space $V_2^{\alpha }\left(Q_T\right)$ for any $f_0(t),$ $f_1(t),$ $b_{10}(t),$ $b_{11}(t),$ $\hspace{0.17em}{b}_{20}(t),$ $b_{21}(t),$ $f(x,t),$ $f_{xx}(x,t)$ $b_1(x,t),$ $b_2(x,t),$ $f_x(0,t),$ $f_x(1,t)$, thus satisfying the conditions of Theorem 1.

For $\lambda =0$, we obtain the boundary value problem in rectangle $Q_T$ for the following linear pseudoparabolic equation:

$$D_{0,t}^{\alpha }v-D_{0,t}^{\alpha }{v}_{xx}-v_{xx}+cu(x,t)=f_{xx}(x,t),$$

(16)

$$\begin{array}{l}{D}_{0,t}^{\alpha }{v}_x(0,t)+v_x(0,t)=0,\\ D_{0,t}^{\alpha }{v}_x(1,t)+v_x(1,t)=0,\end{array}$$

(17)

Theorem 2. 

Let the conditions of Theorem 1 be fulfilled, then, there is a unique regular solution$v\in V_2^{\alpha }\left(Q_T\right).$

Proof. 

We will prove the existence of the solution using the Galerkin method. Using approximate solutions, $v^N(x,t)$, we conducted a search using the following form:

$$v^N(x,t)=\sum _{j=1}^N{C}_{Nj}(t){\Psi }_j(x),$$

(18)

where ${\psi }_j(x)$ are eigenfunctions of the boundary value problem

$$\{\begin{array}{c}{{\psi }^{″}}_j(x)+{\lambda }_j{\psi }_j(x)=0,\\ {{\psi }^{\prime }}_j=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{{\psi }^{\prime }}_j=0.\end{array}$$

It is known [45] that these functions are dense in the space $\stackrel{0}{W_2^1}(0,1)\cap W_2^2(0,1).$ Unknown coefficients $C_{Nj}(t)$ are defined using the system of differential equations, as follows:

$$\begin{array}{l}{\displaystyle \sum _{k=1}^N{\displaystyle \underset{0}{\overset{1}{\int }}\left(D_{0,t}^{\alpha }{C}_{NK}(t)\cdot {\psi }_k{\psi }_j-D_{0,t}^{\alpha }{C}_{NK}(t)\cdot {{\psi }^{″}}_k{\psi }_j-C_{NK}(t)\cdot {{\psi }^{″}}_k{\psi }_j+c\cdot C_{NK}(t)\cdot {\psi }_k{\psi }_j\right)}}dx\\ ={\displaystyle \underset{0}{\overset{1}{\int }}{f}_{xx}(x,t){\psi }_j(x)dx,}\hspace{0.17em}\hspace{0.17em}j=1,\hspace{0.17em}2,\hspace{0.17em}\dots ,\hspace{0.17em}N,\end{array}$$

(19)

with initial conditions

$$v^N(x,0)=v^N(0)=v_0^N={\displaystyle \sum _{i=1}^N{\alpha }_{Ni}\cdot {\psi }_i(x)},$$

(20)

The solution to the linear Cauchy problems (19) and (20) exists for any finite interval $(0,T),$ and for the convergence of the sums (18), it is necessary to establish a priori estimates of solution $u^N$.

Multiply Equation (19) by ${\lambda }_j{C}_{Nj}(t)+\lambda D_{j,t}^{\alpha }{C}_{Nj}(t)+D_{0,t}^{\alpha }{C}_{Nj}(t)$ and use the sum over $j=1,2,\dots ,N$ then, we obtain the following:

$$\begin{array}{l}{\Vert D_{0,t}^{\alpha }{v}^N\Vert }_{2,\Omega }^2+2{\Vert D_{0,t}^{\alpha }{v}_x^N\Vert }_{2,\Omega }^2+{\Vert D_{0,t}^{\alpha }{v}_{xx}^N\Vert }_{2,\Omega }^2+{\Vert v_{xx}^N\Vert }_{2,\Omega }^2\\ +D_{0,t}^{\alpha }\left({\Vert v_{xx}^N\Vert }_{2,\Omega }^2+(1+c){\Vert v_x^N\Vert }_{2,\Omega }^2\right)+c{\Vert v_x^N\Vert }_{2,\Omega }^2\\ \le {\displaystyle \underset{0}{\overset{1}{\int }}{f}_{xx}{D}_{0,t}^{\alpha }{v}^{N}dx-{\displaystyle \underset{0}{\overset{1}{\int }}{f}_{xx}{D}_{0,t}^{\alpha }{v}_{xx}^{N}dx-{\displaystyle \underset{0}{\overset{1}{\int }}{f}_{xx}{v}_{xx}^{N}dx,}}}\end{array}$$

(21)

which yields the following estimation:

$${\Vert v^N(x,t)\Vert }_{V_2^{\alpha }(Q_T)}\le C_5,$$

(22)

The boundedness of the sequence of approximate solutions $\left\{v^N(x,t)\right\}$ follows from estimation (22) in the space $V_2^a{Q}_T$. Since all derivatives included in Equation (19) are quadratically summable over rectangle $Q_T$, we can choose a subsequence $\left\{v^{N_k}\right\}$ and pass it to the limit via $N$ as $k\to \infty$ in system (19). The limit function belongs to the space $Q_T$. Since the system $\left\{{\psi }_j(x)\right\}$ is dense in $L_2(0,1)$, we obtain the limit function $v(x,t)$, and Equation (16) is satisfied almost everywhere in Q_T. Thus, the existence of a regular solution to problems (16), (17), and (15) is proven.

The uniqueness of the solution to problems (16), (17), and (15) is proven using a standard method, via a contradiction; thus, let there be two solutions $v_1(x,t)$ and $v_2(x,t)$. We denote their difference using $v(x,t)=v_1(x,t)-v_2(x,t)$. From Equation (16) and using conditions (17) and (15) we obtain the following:

$$D_{0,t}^{\alpha }v-D_{0,t}^{\alpha }{v}_{xx}-v_{xx}+cv=0,$$

$$D_{0,t}^{\alpha }{v}_x(0,t)+v_x(0,t)=0,$$

$$D_{0,t}^{\alpha }{v}_x(1,t)+v_x(1,t)=0,$$

$$v(x,0)=0.$$

To obtain inequality (21), we find the following:

$${\Vert D_{0,t}^{\alpha }v\Vert }_{2,\Omega }^2+2{\Vert D_{0,t}^{\alpha }{v}_x\Vert }_{2,\Omega }^2+{\Vert D_{0,t}^{\alpha }{v}_{xx}\Vert }_{2,\Omega }^2+{\Vert v_{xx}\Vert }_{2,\Omega }^2+D_{0,t}^{\alpha }\left({\Vert v_{xx}\Vert }_{2,\Omega }^2+(1+c){\Vert v_x\Vert }_{2,\Omega }^2\right)+c{\Vert v_x\Vert }_{2,\Omega }^2\le 0,$$

which implies ${\Vert v_{xx}\Vert }_{2,\Omega }^2+(1+c){\Vert v_x\Vert }_{2,\Omega }^2\le 0$, then $v(x,t)=0$, so, $v_1(x,t)=v_2(x,t)$ in $Q_T$.

Thus, Theorem 2 is completely proven. □

From estimates (11) and (12), it follows that set $A_{\lambda }$ is closed, and from Theorem 2, it follows that $0\in A_{\lambda }$. Now, let us show that $A_{\lambda }$ is an open set. Let problems (13)–(15) be solvable for $\lambda ={\lambda }_0$. We show that it is solvable for $\lambda ={\lambda }_0+\overline{\lambda }$, where $\overline{\lambda }>0$, as follows:

$$\begin{array}{l}{L}_{{\lambda }_0}v\equiv D_{0,t}^{\alpha }v-D_{0,t}^{\alpha }{v}_{xx}-v_{xx}+cv(x,t)-{\lambda }_0\left[b_{1xx}(x,t)D_{0,t}^{\alpha }v(0,t)+b_{2xx}(x,t)v(0,t)\right]\\ =f_{xx}(x,t)+\overline{\lambda }\left[b_{1xx}(x,t)D_{0,t}^{\alpha }v(0,t)+b_{2xx}(x,t)v(0,t)\right]=F(x,t),\end{array}$$

(23)

$$\begin{array}{l}{D}_{0,t}^{\alpha }{v}_x(0,t)+v_x(0,t)+{\lambda }_0\left[f_0(t)+b_{10}(t)D_{0,t}^{\alpha }v(0,t)+b_{20}(t)v(0,t)\right]\\ =\overline{\lambda }\left[f_0(t)+b_{10}(t)D_{0,t}^{\alpha }v(0,t)+b_{20}(t)v(0,t)\right]={\Phi }_1(t),\\ \\ D_{0,t}^{\alpha }{v}_x(0,t)+v_x(0,t)+{\lambda }_0\left[f_1(t)+b_{11}(t)D_{0,t}^{\alpha }v(0,t)+b_{21}(t)v(1,t)\right]\\ =\overline{\lambda }\left[f_1(t)+b_{11}(t)D_{0,t}^{\alpha }v(0,t)+b_{21}(t)v(0,t)\right]={\Phi }_2(t),\end{array}$$

(24)

Take an arbitrary function $\omega (x,t)\in V^{\alpha }(Q_T)$, and substitute it into the right part of (23) and (24).

We denote the following:

$$\begin{array}{l}B=\{v:\hspace{0.17em}v\in L_{\infty }(0,T;\hspace{0.17em}{L}_2(\Omega )\cap W_2^1(\Omega )),\hspace{0.17em}{D}_{0,t}^{\alpha }v\in L_{\infty }(0,T;\hspace{0.17em}{L}_{\infty }(\Omega )\cap W_2^1(\Omega )),\hspace{0.17em}v(x,0)=0,\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\Vert v\Vert }_B\equiv {\Vert v\Vert }_{L_{\infty }(0,T;\hspace{0.17em}{L}_2(\Omega )\cap W_2^1(\Omega ))}+{\Vert D_{0,t}^{\alpha }v\Vert }_{L_{\infty }(0,T;\hspace{0.17em}{L}_2(\Omega )\cap W_2^1(\Omega ))}\le R\},\end{array}$$

$$B_1=\left\{U:\hspace{0.17em}{\Vert U\Vert }_{V_2^{\alpha }(Q_T)}\le R,\hspace{0.17em}\hspace{0.17em}U(x,0)=0\right\};$$

where $B_1\subset B$, $R$ is related to the constant from (12) (i.e.,$2c\le R$).

$$F(x,t)\equiv f_{xx}(x,t)+\overline{\lambda }\left[b_{1xx}(x,t)D_{0,t}^{\alpha }\omega (o,t)+b_{2xx}(x,t)\omega (0,t)\right]\in L_2(Q_T),$$

$${\Phi }_1(t)\equiv \overline{\lambda }\left[f_0(t)+b_{10}(t)D_{0,t}^{\alpha }\omega (0,t)+b_{20}(t)\omega (0,t)\right]\in L_2(Q_T),$$

$${\Phi }_2(t)\equiv \overline{\lambda }\left[f_1(t)+b_{11}(t)D_{0,t}^{\alpha }\omega (0,t)+b_{21}(t)\omega (0,t)\right]\in L_2(Q_T).$$

As ${\lambda }_0\in A_{\lambda }$, there exists a unique solution to problems (23), (24), and (15), we can assume that $v=K(\omega )\in V_2^{\alpha }(Q_T)$. Moreover, (11) and (12) yields the following:

$${\Vert v\Vert }_{V_2^{\alpha }(Q_T)}={\Vert K(\omega )\Vert }_{V_2^{\alpha }\left(Q_T\right)}\hspace{0.17em}\hspace{0.17em}\le C+C_1(R,\hspace{0.17em}\overline{\lambda },\hspace{0.17em}T)\le R.$$

(25)

We choose $\overline{\lambda }$ so that $C_1\left(R,\hspace{0.17em}\overline{\lambda },\hspace{0.17em}T\right)\le C.$

Let us show that the operator $K$ is compressive. To do this, consider the following solutions $v_1,\hspace{0.17em}{v}_2\in B_1:\hspace{0.17em}\hspace{0.17em}v=v_1-v_2$, $v=K\left({\omega }_1\right)-K\left({\omega }_2\right)$, ${\omega }_1,\hspace{0.17em}\hspace{0.17em}{\omega }_2\in B,\hspace{0.17em}\hspace{0.17em}\omega ={\omega }_1-{\omega }_2$,

$$\begin{array}{l}S\omega \equiv D_{0,t}^{\alpha }v-D_{0,t}^{\alpha }{v}_{xx}-v_{xx}+cv(x,t)-{\lambda }_0\left[b_{1xx}(x,t)D_{0,t}^{\alpha }v(0,t)+b_{2xx}(x,t)v(0,t)\right]\\ =\stackrel{-}{\lambda }\left[b_{1xx}(x,t)D_{0,t}^{\alpha }\omega (0,t)+b_{2xx}(x,t)\omega (0,t)\right]\end{array}$$

Regarding the derivation of estimates (11) and (12), the following can be shown:

$${\Vert v\Vert }_{V_2^{\alpha }\left(Q_T\right)}=\overline{\lambda }C{\Vert \omega \Vert }_B,$$

Choosing $\overline{\lambda }>0$, with a low enough value, the following inequality is obtained: $q=\overline{\lambda }C<1$,

$${\Vert v\Vert }_{V_2^{\alpha }\left(Q_T\right)}={\Vert v_1-v_2\Vert }_{V_2^{\alpha }\left(Q_T\right)}={\Vert K{\omega }_1-K{\omega }_2\Vert }_{V_2^{\alpha }\left(Q_T\right)}\le q{\Vert {\omega }_1-{\omega }_2\Vert }_B.$$

(26)

Then, as per (25) and (26), it follows that there exists $\overline{\lambda }>0$ such that equation $v=K\omega$ is unambiguously solvable and the element $v_0\in B_1\hspace{0.17em}\Rightarrow \hspace{0.17em}{v}_0=K{\omega }_0$ is obtained. It is clear that this element will satisfy $L_{{\lambda }_0{v}_0}=F$ and conditions (24) and (15) (i.e., $A_{\lambda }$ is an open set). Then, according to the theorem concerning the continuation method with parameters [48], the solvability of problems (5)–(7) follows for $\lambda =1$.